Question: Simplify and expand the following expression: $ \dfrac{n}{n - 6}+\dfrac{2n - 8}{3n + 4} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(n - 6)(3n + 4)$ Multiply the first term by $\dfrac{3n + 4}{3n + 4}$ $ \begin{align*} \dfrac{n}{n - 6} \times \dfrac{3n + 4}{3n + 4} & = \dfrac{(n)(3n + 4)}{(n - 6)(3n + 4)} \\ & = \dfrac{3n^2 + 4n}{(n - 6)(3n + 4)}\end{align*} $ Multiply the second term by $\dfrac{n - 6}{n - 6}$ $ \begin{align*} \dfrac{2n - 8}{3n + 4} \times \dfrac{n - 6}{n - 6} & = \dfrac{(2n - 8)(n - 6)}{(3n + 4)(n - 6)} \\ & = \dfrac{2n^2 - 20n + 48}{(3n + 4)(n - 6)}\end{align*} $ Now we have: $ = \dfrac{3n^2 + 4n}{(n - 6)(3n + 4)} + \dfrac{2n^2 - 20n + 48}{(3n + 4)(n - 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3n^2 + 4n + 2n^2 - 20n + 48}{(n - 6)(3n + 4)} $ $ = \dfrac{5n^2 - 16n + 48}{(n - 6)(3n + 4)}$ Expand the denominator: $ = \dfrac{5n^2 - 16n + 48}{3n^2 - 14n - 24}$